So, I recently acquired a Larry Bird autograph from the 1992 Classic World Class Athletes multisport set. The back of the card says "Congratulations, you have just received this extremely limited classic autograph card."

So I know it's a legit auto. My problem is that I am having a very hard time finding any examples of this cards existence anywhere. The Beckett online checklist doesn't list this card as part of the set, but I've read the wording of the set "several athletes have randomly inserted autographs into the set"... or words to that effect...

Anyway, I'm trying to figure out how many of these were printed and what this might be worth. I've been digging for almost a full day now and can't find much to go on.

I believe this is the non-autographed version of the card you are asking about.
I know the cards came as a complete 60 card set, as shown below.
However the sealed sets don't say anything on the front or back about random autographed versions. So I am not sure what product these autographs would have come in. They made 295,000 of these "limited edition" sets, so it could be possible that they made another version that had a random autograph in it. I know that Classic made a ton of cards in the early 90s and occasionally would take older cards (or use the printing plates to re-run the front, and add their standard "congratulations autograph card" back) and have them autographed and inserted into later product. So it would be possible that they had some of these cards inserted into other 1992 or later product. What is the copyright date on the back of the card?

Ok, for some reason some of the older Classic cards don't have dates on the back, while others do. I'm guessing your card's back looks like the following:
There is also a Muhammad Ali autograph version on eBay right now, which was graded by BGS. So if your card has the same back as that then it has a good chance of being legit. Unfortunately, I still can't find much information regarding where these autographs were inserted or how they were originally available.

I can't find it either... I'm wondering if I should send it to get graded... That would solidify it's legitimacy, wouldn't it?

I'm wondering how it could be a fake though, being as it is written on a card that has the Classic auto message on the back of the card..
Also, one thing about that Ali... That is one of the three cards that is included in the Beckett.com checklist for that set. It specifically says that it's autographed...

It has a Capriati, an Ali and a Pete Sampras... I wonder if they only add to their online checklist if they can confirm the card's existence?

Either way, if I send it in to be graded, I should get a definitive answer...

I am actually now intrigued to find out how the autographed versions were originally available. In doing my research, I have found copies of the Capriati and Ali autographs, but not much else (aside from your Larry Bird). However no one that owns one seems to know where they originated from.
The card could easily be faked since there is no copyright or trademark indication on it. The back is pretty basic and the front could be done pretty easily. The big upside is you don't see many of these cards, which is usually a good indication that they weren't faked. Sending it in for grading or authenticating is a pretty good idea to be certain, but I wouldn't worry too much about that unless a few of them started popping up all of a sudden.